Green and Kubo forge the arrow of time

Transport theory describes the response of a macroscopic current to a thermodynamic force, thus producing entropy and apparently violating time-reversal symmetry. In this note I report a pedagogical derivation of the Green-Kubo formula for transport coefficients that highlights the intrinsically dynamical nature of this formula and showcases the relation between the apparent breach of time-reversal symmetry and the non-commutativity of the low-frequency / low-wavevector limits of the conserved-density susceptibilities, from which the formula can be established.